Whakaoti i te x^2+6x-72=0

$$x ^ { 2 } + 6 x - 72 = 0$$

Whakaoti mō x

$$x=-12$$

$$x=6$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}+6x-72 mā te whakamahi i te tātai x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=6 ab=-72$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -72.

$$-1,72 -2,36 -3,24 -4,18 -6,12 -8,9$$

Tātaihia te tapeke mō ia takirua.

$$-1+72=71 -2+36=34 -3+24=21 -4+18=14 -6+12=6 -8+9=1$$

Ko te otinga te takirua ka hoatu i te tapeke 6.

$$a=-6 b=12$$

Me tuhi anō te kīanga whakatauwehe \left(x+a\right)\left(x+b\right) mā ngā uara i tātaihia.

$$\left(x-6\right)\left(x+12\right)$$

Hei kimi otinga whārite, me whakaoti te x-6=0 me te x+12=0.

$$x=6 x=-12$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-72. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=6 ab=1\left(-72\right)=-72$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -72.

$$-1,72 -2,36 -3,24 -4,18 -6,12 -8,9$$

Tātaihia te tapeke mō ia takirua.

$$-1+72=71 -2+36=34 -3+24=21 -4+18=14 -6+12=6 -8+9=1$$

Ko te otinga te takirua ka hoatu i te tapeke 6.

$$a=-6 b=12$$

Tuhia anō te x^{2}+6x-72 hei \left(x^{2}-6x\right)+\left(12x-72\right).

$$\left(x^{2}-6x\right)+\left(12x-72\right)$$

Tauwehea te x i te tuatahi me te 12 i te rōpū tuarua.

$$x\left(x-6\right)+12\left(x-6\right)$$

Whakatauwehea atu te kīanga pātahi x-6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-6\right)\left(x+12\right)$$

Hei kimi otinga whārite, me whakaoti te x-6=0 me te x+12=0.

$$x=6 x=-12$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$x^{2}+6x-72=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 6 mō b, me -72 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-72\right)}}{2}$$

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\left(-72\right)}}{2}$$

Whakareatia -4 ki te -72.

$$x=\frac{-6±\sqrt{36+288}}{2}$$

Tāpiri 36 ki te 288.

$$x=\frac{-6±\sqrt{324}}{2}$$

Tuhia te pūtakerua o te 324.

$$x=\frac{-6±18}{2}$$

Nā, me whakaoti te whārite x=\frac{-6±18}{2} ina he tāpiri te ±. Tāpiri -6 ki te 18.

$$x=\frac{12}{2}$$

Whakawehe 12 ki te 2.

$$x=6$$

Nā, me whakaoti te whārite x=\frac{-6±18}{2} ina he tango te ±. Tango 18 mai i -6.

$$x=-\frac{24}{2}$$

Whakawehe -24 ki te 2.

$$x=-12$$

Kua oti te whārite te whakatau.

$$x=6 x=-12$$

Ngā Upane mō te Whakaoti i te Pūrua

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$x^{2}+6x-72=0$$

Me tāpiri 72 ki ngā taha e rua o te whārite.

$$x^{2}+6x-72-\left(-72\right)=-\left(-72\right)$$

Mā te tango i te -72 i a ia ake anō ka toe ko te 0.

$$x^{2}+6x=-\left(-72\right)$$

Tango -72 mai i 0.

$$x^{2}+6x=72$$

Whakawehea te 6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 3. Nā, tāpiria te pūrua o te 3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+6x+3^{2}=72+3^{2}$$

Pūrua 3.

$$x^{2}+6x+9=72+9$$

Tāpiri 72 ki te 9.

$$x^{2}+6x+9=81$$

Tauwehea x^{2}+6x+9. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+3\right)^{2}=81$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{81}$$

Whakarūnātia.

$$x+3=9 x+3=-9$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$x=6 x=-12$$